Kernel of a characteristic action on an abelian group implies strongly image-potentially characteristic
This article gives the statement and possibly, proof, of an implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., kernel of a characteristic action on an abelian group) must also satisfy the second subgroup property (i.e., strongly image-potentially characteristic subgroup)
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Statement with symbols
Then, there exists a group with a surjective homomorphism such that both the kernel of and are characteristic subgroups of .
Given: , an abelian group , a homomorphism with kernel . is characteristic in .
To prove: There exists a group with a surjective homomorphism such that both the kernel of and are characteristic subgroups of .
Proof: Let and be the quotient map.
- By assumption, , the kernel of is characteristic in .
- is characteristic in : This follows from fact (1).
- : Since is abelian, the quotient acts on by conjugation (fact (2)). In other words, any two elements in the same coset of act the same way on by conjugation. Thus, a coset of centralizes if and only if the element of in the coset centralizes , which happens if and only if the element is in . Thus, (because the action is trivial) and is equal to .
By step (1), the kernel of is characteristic in . Combining steps (2) and (3) yields that is also characteristic in , completing the proof.